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Time-Dependent Covariates “Survival” More in PROC PHREG Fengying Xue,Sanofi R&D, China Michael Lai, Sanofi R&D, China

ABSTRACT Survival analysis is a powerful tool with much strength, especially the semi-parametric analysis of COX model in PHREG, the most popular one. How to explain its enormous popularity? The most important reason is that it does not require you to choose some particular probability distribution under the proportional-hazards assumption. While in some cases, such as in long-term follow-up study or some covariates whose attribute really change over process (such as age, salary), the proportional-hazards assumption of constant hazard ratio is frequently violated, and PHREG can also make it. This is the second reason; it is relatively easy to incorporate time-dependent covariates. It provides the chance to modulate dynamic design, leading to a more robust and accurate outcome.

INTRODUCTION We begin by defining a time-dependent variable and use Stanford heart transplant study as example. We also state the general formula for Cox model and how the Cox proportional hazards (PH) model can be extended to allow time-dependent variables, followed by a discussion bases on Stanford heart transplant study, including a description of the hazard ratio, two methods to handle time-dependent variable in PHREG. At last, we will check PH assumption by using multiple methods for accuracy and robustness.

BRIEF BACKGROUND Time-dependent covariates are those that may change in value for a given subject over the course of observation. In contrast, time-independent covariates are those whose value remains constant over time. Take Stanford heart transplant study as example, regarding covariate TRANS (which is equal to 1 if the patient has already had a transplant at time t and 0 otherwise). In this example, we see that TRANS is time-dependent variable whose value changed after transplanted at time t0 for the transplant one. While for the non- transplant one, its value is always 0. All in all, TRANS is a time-dependent variable for this study.

      

Based on attribute and reason of the value change, time-dependent variables can be classified into two categories: Internal and Ancillary. For internal variable, whose value change because of “internal” characteristics or behavior of the individual while ancillary variable whose values change because of “external” characteristics. An example of an ancillary variable is air pollution index at time t for a particular geographical area. Additionally, for above Stanford heart transplant, TRANS, it is part internal and part ancillary variable, individual traits, eligible transplant criteria, is internal and available donor is ancillary.  

Why time-dependent analysis is so powerful in survival analysis? Firstly, since that outcome or endpoint (time to event) is a time related variable, if the explanatory variable is time-dependent variable it is probability that it is heavily correlated with each other. In such instances, it is easy to confuse the “causation” of effect and misleading the result. Secondly, some variables that are important to the risk of an endpoint will vary in individuals over the course of a study and there is no way to control them (i.e. keep them constant) by controlling the people’s risk factors, such as people’s alcohol related drinking habits, meat intake, and dairy intake. Additionally, FDA has asked for clarifying for “what estimated”, also citing “De facto” estimate as their interest, so we must be aware of this kind of variables, time-dependent. If not, treat dependent as independent, it may cause bias in the estimation, even more incorrect inference regardless of significance of effects, and it may over fit model and cost much extraneous time and without estimate improvement. So let’s extent PH COX model to extended COX model, time-depend COX model.

 

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 MISLEADING OUTCOME There is one misleading outcome which not takes time-dependent into consideration, treat TRANS as time-indepent variable. The traditional method is using PH COX model, 1 ∗ 2 ∗3 ∗ to estimate the COX regression of SURV1 on trasplant status(TRANS), controlling for SURG and

AGEACCPT, TRANS and SURG are all treated as time-independent variables.  

  The result as below output show very strong effects on both transplant status and age at acceptance. We saw that each additionally year of age at the time of acceptance is associated with a 6 percent increase in the hazard of death. On the other hand, the hazard for those who received transplantation is only about 18 percent of the hazard for those who do not. Or equivalenetly, those who did not receive transplans are about 4.5(1/0.181-1≈4.5) times more likely to die at any given timepoint.  

  Perhaps the age effect maybe real, but the transplantation effect is almost surely an artifact. The main reason is that TRANS is actually consequence of the dependent variable: an early death prevent a patient from getting transplantation, or taking the reciprocal, the more time patient live, the more chance to receive transplantation, so it was difficult to determine if transplantation actually reduced the risk of death or if people who lived longer were more likely to receive a transplantation. Below plot shows the primary reason for this misleading model. For transplant patients, transplant win time should contribute to TRANS effect not the total survival time. In this analysis, the total survival will be favor for transplant, and then enlarge transplant effect. In fact, wait time belong to non-transplant effect the same as for non-transplant, so it will be better to split total duration of transplant patients into two ones, one is for wait time period without transplantation, another is transplant win time with transplantation. While for non-transplant patients, they are only with wait-time period. Under this split, no matter Transplant or Non-transplant, they will be treated fairly. It will make the result more accurate. This is also the primary idea for time dependent method, split into many intervals, during which hazard is constant, and then give analysis on each intervals, and combine them give a final outcome, generally. censored or death wait-time transplant win time Transplant Received transplant Total survival time   censored or death wait-time Non-Transplant Not received transplant Total survival time

 

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Under this model, it indicates that transplantation has no effect on the hazard of death. The effect of age at acceptance is somewhat smaller than before, it stationary relatively. It is different deeply from the previous traditionally one. This one is more accurate and more close the truth.

  PROGRAMMING STATEMENT  Programming statement is used to create or modify the values of the exlanaroty variables in the model statement. It is especially usefull in fitting models with time-dependent coavariabtes. Its syntax likes in data-step. Now let’s see how we can use SAS to create a time-dependent covariate that changes for each individual from 0 to 1 at the time of the transplantation.

 

  Perhaps there are two mysteries. The first is that a new variable is created, called ‘TRANS’ which equals 0 if the Waiting time is greater or equal than the survival time or if the waiting time is missing, otherwise TRANS=1. Does this make intuitive sense? The second is the result is magic, they are exactly the same with the result of count process. For the first question, the answer should be ‘no’, not intuitive, but it is correct. It will take a little effort to understand what this conditional statement in PHREG is doing. The point is that the conditional statement in PHREG is handled differently than in a data step. If they are the same, programming statement can’t be time-dependent data carrier. It’s more complicated than simply evaluating whether WAIT>=SURV1 for each patient.

      Let’s use a plot to see its process clearly, red stars stand for survial starting or ending points, black stars stand for event time points and red heart stands for transplantation.

SURV1 is a variable that is evaluated at each event time point, a comparison is made between the event time and each patient’s wait time. Based on those comparisons, each patient’s duration was spited into multiply intervals, and the TRANS value will be 0 or 1, time-depend variable.

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censored or death Wait-time transplant win time int1 int2 …. Int i int (i+1) …… int j Transplant Pt1 Received transplant Total survival time       censored or death Wait-time int1 int2 …. Int k ….. int l Not Transplant Pt2 Received transplant Total survival time Its corrsponding tables as below, suppose Pt1 is transplant patient while Pt2 is not. Transplant:

interval start end condition trans Int 1 0 Event 1 Wait>Event 1 0 Int 2 Event 1 Event 2 Wait>Event 2 0 …. …. …. …… …. Int i Event i-1 DOT Wait=DOT 0 Int i+1 DOT Event i Wait<Event i+1 1 …. …. …. …… …. Int j Event j-1 SURV1 Wait<Event j 1

Not Transplant:

interval start end condition trans Int 1 0 Event 1 Wait>Event 1 0 Int 2 Event 1 Event 2 Wait>Event 2 0 Int 3 Event 3 Event 3 Wait>Event 3 0 …. …. …. …… …. Int l Event l-1 SURV1 Wait> Event l 0

From above plots and tables we find that programming statement is the same as count process in fact, the difference is that programming statement split more interverls, if we combine all intervals by TRANS staus, we will got the same intervals as count process, so that’s not surprise we got the same result from the two method. There is another confusion, why assign 0.1 for patients who are with WAIT=0 or SURV1=0? Below are three special patients. ID DOA DOT DLS DEAD DUR SURG TRANS WARIT SURV1 3 01/06/1968 01/06/1968 01/21/1968 1 16 0 1 0 15

45 01/05/1971 01/05/1971 02/18/1971 1 45 0 1 0 44

15 09/27/1968 09/27/1968 1 1 1 0 10000 0 Patient, ID=15, who was dead once he was accepted into this study, so survial time is 0. If we treated its interval as (0, 0], this patients will be excluded from analysis in counting process. But for him, even for this study, 0 is an event time, it will lead to L1 was missed in below partial likelihood funciton. By the way, the minxium event time except this patient, is 1, so that we can assign a value, called µ , μ ⋲(0, 1), such as 0.1, it will give the chance for patient ID=15 to be an element of partial likelihood function and this is the fact. ⋯ , Where j is the event sequence by survival time. 

 While for another two patients who took transplantation once they accepted into the study, and they were at risk or not during (0, µ], there are two and only two possible.

(1) if we suppose their wait time ≥ µ, they were not transplanted during (0, µ], TRANS=0 (2) if we suppose their wait time <µ, they were transplanted during (0, µ], TRANS=1

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In mathematical logic, any of them is possible, we are not sure. The two will lead to different results. For (1), it is the same as the previeous example, as assign 0.1 for all three cases. It is caused by SURV1 & WAIT definition (before), without plus 1, so 0 appeared as survival time or wait time. If we use Define (after), plus one day for SURV1 and WAIT, it will avoid this issue and get the same results. It is because that its assessment is the same regardless based on value or ranked value due to the partial likelihood function, baseline hazard function is canceled out, depend only on the ranks of the event times not their numerical values. This implies that any monotonic transformation of the event times will leave the coefficient estimates unchanged. So, WAIT=0 and SURV1=0, with same ranked value, assign= µ ,such as 0.1. This is the whole history.

Variable Define(before) Define(after) SURV1 DLS-DOA DLS-DOA+1 WAIT DOT-DOA DOT-DOA+1

As shown, the counting process requires substantially more coding. However, this upfront effort makes it easier to detect and correct errors because a data set is created and can be debugged. The programming statements are faster to code and can handle special case, but the coding seems to be tricky and there is no way to detect if the time-varying covariates have been coded correctly. Furthermore, when using the programming statement method, a temporary dataset containing the time-varying covariates has to be created each time PROC PHREG is run. Depending on how large the dataset is, this could drastically increase computing time. For these reasons, we prefer the counting process. We also more prefer to use one to validate another one for robustness. Our results from the examples illustrated how impactful time-dependent variables can be in Cox PH model. When we used static variables only in the model, TRANS had effect on SURV1. But when we used more time points of TRANS status, we saw a very insignificant effect of TRANS on SURV1. So it is strengthful to use time-dependent variables if it is, meanwhile it should be caution to use time-dependent modle since that if not it will cause much time in running model and cost much time to investigate and demonstrate it. So before we use time-dependent model, we must test its PH assumption and search for clinical expert suggestions.  ASSESS PH ASSUMPTION  There are three general approaches for assessing the PH assumption, again listed here, graphical, goodness of fit and time-dependent variables

GRAPHICAL

We now briefly overview each approach, starting with graphical techniques. Totally, 2 types of plots are involved. The quick and easy one is the Kaplan-Meier plot which estimates of the survival function to checks of proportional hazards. If proportional hazards hold, the graphs of the surival function should look "parallel", in the sense that they should have basically the same shape, should not cross, and should start close and then diverge slowly through follow up time.  

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There is another type of graphical technique available. The mose popular one, it involves comparing estimated ln(–ln) survivor curves over different (combinations of) categories of variables being investigated. Based on below calculation, Parallel curves indicate that the PH assumption is satisfied.

 

 

   

From plot methods, it is easy to get an idea, but the big issue is that how ‘Parallel’ is parallel? From above two kinds of plots, it seems that we can find any time-dependent clue. But how ‘Parallel’ is Parallel? We would like to find one value with statistical meaning. Beyond this method, we have alternative method to test PH assumption.

GOODNESS OF FIT

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A famous method for evaluating PH assumption is to examine the Schoenfeld residuals. The Schoenfeld residual is defined as the difference between covariate for observation and the weighted average of the covariate values for all subjects still at risk when observation experiences the event.

For covariate xk, it is the difference between the ith individual’s covariate, xik, and the “expected” value of xk for all people at risk, where pj is the probability that person j had event at time ti. Schoenfeld residuals are mainly used to detect departures from the proportional hazards assumption. If there is a pattern in these residuals against survival time, the PH assumption is questionable. If PH satisfied, plots will show no trend over time and its average is 0 across time.

Additionly, we can also take expected function, It can be shown that:

Where βk(ti) is a time dependent coefficient (or, alternatively, the corresponding covariate is time dependent). We also assess PH from β coefficent.

Firstly, we would like to use residual plot to get a rough idea. This can be attrived by Proc PHREG to get Schoenfeld residual.

And then get its residual trend with according survival time,

 

 

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From above plot, we find that lack of residuals for long survival times for the ‘trans’ covariate as in red ellipses, it indicates a possible time dependent coefficient. As it turns out, a time dependent covariate analysis was required for this dataset. This is just a plot impression, we would like more to see its relationship, since that the idea behind the statistical test is that if the PH assumption holds for a particular covariate then the Schoenfeld residuals for that covariate will not be related to survival time. So it is better to get its relation coefficient and its P value to test significance or not.

And the result is as below.

And the result is as below,

From Schoenfeld residual plot and its rank test, we see that Schoenfeld residual correlation with survival time, PH assumption violated significantly at P=0.01 level.

There is another good of fitness method for PH assessment via Proc PHREG also. The procedure, developed by Lin, Wei, and Zing (1990), can detect violations of proportional hazards by using a transform of the Schoenfeld residuals known as the empirical score process. The empirical score process under the null hypothesis of no model misspecification can be approximated by zero mean Guassian processes, and the observed score process can be compared to the simulated processes to asses departure from proportional hazards.

It is easy to execute, adding ph options on the assess statement. All the exploratory variables will be assessed this way. We also specify the resample option, which performs a supremum test of the null hypothesis that PH assumption holds. Essentially, the supremum tests calculate the proportion of 1000 simulations that contain a maximum standardized score larger than the observed maximum standardized score. This proportion is reported as the p-value. If only a small proportion, say 0.05, of the simulations have a standardized score larger than the observed maximum, then that suggests that violation of proportional hazard.

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Let’s look at the model with just a linear effect for TRANS.

We find that the solid lines represent the observed standardized score, while dotted lines represent 20 simulated sets of standardized score under the null hypothesis that PH assumption holds. A solid line that falls significantly outside the boundaries set up collectively by the dotted lines suggest that observed standardized score do not conform to the expected standardized score. This graph look particularly alarming and the supremum tests are significant, suggesting that PH assumption violated.

TIME-DEPENDENT VARIABLES

There is still another method to test PH assumption, it is just like time-dependent method, putting time related (suppose) variable into PH COX model to test its significient use -2 L test. This is simple but with more clinical science and so without additionally specify in this paper.

CLONCLUSION

In this paper we introduce extended COX model via PH COX model and then have shown an example with two methods used to handle time-varying covariates, count process and programming statement. We prefer the counting process. We prefer to use one to validate another one for robustness. It is strengthful to use time-dependent variables if it is, meanwhile it should be caution to use time-dependent model since that if not it will cause much time in running model and cost much time to investigate and demonstrate it. So before we use time-dependent model, we must test it PH assumption and search for clinical expert suggestion. We also introduce some methods to check PH assumption, it is important, and will favor an object outcome. So before using time-dependent model, we must test PH assumption and ask for expert’s suggestion in case of over fitting model or under estimate model. Additionly, they can not do without PROC PHREG, a very useful tool. 

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REFERENCE

M. Gail, K. Krickeberg, J.M. Samet, A. Tsiatis, W. Wong(2012) Survival Analysis A Self-Learning Text Third Edition Allison, P. D. (2010) Survival Analysis Using SAS: A Practical Guide: Sas Inst Teresa M. Powell, Melissa E. Bagnell(2012), Your “Survival” Guide to Using Time‐Dependent Covariates Mark Jones (2016), Time-dependent bias in observational studies of oseltamivir SAS Seminar (2014) Introduction to Survival Analysis in SAS CONTACT INFORMATION

Your comments and questions are valued and encouraged. Contact the author at: Name: Fengying Xue Enterprise: Sanofi R&D China Address: 2th Floor, HNA Plaza, No 108, Jianguo Road, Chaoyang District City, State ZIP: Beijing, 100022 Work Phone: 86 10 6563 4915 E-mail: Fengying.xue@sanofi.com Name: Michael Lai Enterprise: Sanofi R&D China Address: 2th Floor, HNA Plaza, No 108, Jianguo Road, Chaoyang District City, State ZIP: Beijing, 100022 Work Phone: 86 10 6563 4900 E-mail: Michael.Lai@sanofi.com